This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to the problems: 3. Rectangular Shamba Let the width of the rectangular shamba be x meters. a) The length: The length is 2 meters longer than the width. L = x + 2 The expression for the length is x+2. b) The area: The area of a rectangle is given by length × width. A = L × x Substitute the expression for L: A = (x+2)x The expression for the area is x(x+2). (Note: The problem states the area is 48 m², so x(x+2) = 48, but the question asks for an expression for the area, not to solve for x.) 4. Farmer distributing rice Step 1: Identify the total amount of rice and the capacity of each bag. Total rice = 5000 kg Rice per bag = 12.5 kg Step 2: Determine the number of bags needed. The number of bags is the total rice divided by the rice per bag. Number of bags = Total riceRice per bag Number of bags = (5000)/(12.5) Step 3: Use the reciprocal of 12.5 as requested. The reciprocal of 12.5 is (1)/(12.5). Number of bags = 5000 × (1)/(12.5) Step 4: Calculate the reciprocal of 12.5. (1)/(12.5) = (1)/(25)2 = (2)/(25) = 0.08 Step 5: Multiply the total rice by the reciprocal. Number of bags = 5000 × 0.08 Number of bags = 400 The total number of bags needed is 400. 5. Medicine concentration Step 1: Write the given concentration. Concentration C = 0.0006214 g/mL. Step 2: Express the concentration in scientific notation. Move the decimal point 4 places to the right to get a number between 1 and 10. C = 6.214 × 10^-4 g/mL Step 3: Express the concentration in the form 10^n. We need to find n such that 10^n = 0.0006214. This means n = _10(0.0006214). Using the scientific notation: n = _10(6.214 × 10^-4) Apply the logarithm property (AB) = A + B: n = _10(6.214) + _10(10^-4) Apply the logarithm property (A^B) = B A: n = _10(6.214) - 4 Step 4: Find _10(6.214) using logarithm tables. From logarithm tables: _10(6.21) = 0.7931 The mean difference for 4 (in the row for 62) is 3. So, _10(6.214) = 0.7931 + 0.0003 = 0.7934. Step 5: Substitute the value back into the equation for n. n = 0.7934 - 4 n = -3.2066 Step 6: Write the concentration in the form 10^n. C = 10^-3.2066 g/mL The concentration expressed in the form 10^n is 10^-3.2066.